A PARAMETRIC BOOTSTRAP FOR HEAVY-TAILED DISTRIBUTIONS

It is known that Efron’s bootstrap of the mean of a distribution in the domain of attraction of the stable laws with infinite variance is not consistent, in the sense that the limiting distribution of the bootstrap mean is not the same as the limiting distribution of the mean from the real sample. Moreover, the limiting bootstrap distribution is random and unknown. The conventional remedy for this problem, at least asymptotically, is either the m out of n bootstrap or subsampling. However, we show that both these procedures can be unreliable in other than very large samples. We introduce a parametric bootstrap that overcomes the failure of Efron’s bootstrap and performs better than the m out of n bootstrap and subsampling. The quality of inference based on the parametric bootstrap is examined in a simulation study, and is found to be satisfactory with heavy-tailed distributions unless the tail index is close to 1 and the distribution is heavily skewed.

A survey of limit laws for bootstrapped sums

Concentrating mainly on independent and identically distributed (i.i.d.) real-valued parent sequences, we give an overview of first-order limit theorems available for bootstrapped sample sums for Efron's bootstrap. As a light unifying theme, we expose by elementary means the relationship between corresponding conditional and unconditional bootstrap limit laws. Some open problems are also posed.